The Density Hypothesis

The Density Hypothesis is the assertion

for all . Note that this is nontrivial
only when
.

The Density Hypothesis follows from the Lindelöf Hypothesis.
The importance of the Density Hypothesis is that, in terms of
bounding the gaps between consecutive primes, the density hypothesis
appears to be as strong as the Riemann Hypothesis.

Results on are generally obtained from mean
values of the zeta-function. Further progress in this
direction, particularly for close to , appears
to be hampered by the great difficulty in estimating the
moments of the zeta-function on the critical line.